Working with 1 carrot at a time, cut carrots crosswise into 2- to 3-inch pieces. Cut off a thin slice lengthwise from each carrot piece to create a flat edge. Lay the carrot flat on a cutting board. Cut lengthwise into thin (about ⅛-inch-thick) planks. Stack a few of the planks together, and slice them lengthwise into thin (about ⅛-inch-thick) strips.
Whisk together the cooking oil, sesame paste, soy sauce, vinegar, sesame oil, ginger, garlic, honey, and 3 tablespoons of the spicy chile crisp in a large bowl. The sauce should be thin and pourable; whisk in water, 1 tablespoon at a time, if needed to thin. Set aside.
Cook the noodles according to package directions, adding carrots and spinach during the final 30 seconds of cooking time. Drain well, and add to sauce in the bowl; toss to coat. (Dressed noodles can be served warm, chilled, or at room temperature.)
To serve, drizzle noodle mixture with remaining 2 tablespoons chile crisp. Sprinkle with scallions, and garnish with sesame seeds.